An improved bound on the number of dot products determined by a finite point set in the plane

TL;DR

This paper improves the lower bound on the number of distinct dot products determined by a finite point set in the plane, advancing understanding of geometric combinatorics.

Contribution

It provides a tighter lower bound on the number of distinct dot products, building on prior work to refine the exponent.

Findings

Established a new lower bound: |{p·q}| ≳ |P|^{2/3 + 7/1425}

Improved upon previous bounds in geometric combinatorics

Enhances understanding of dot product distributions in finite point sets

Abstract

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the improved lower bound \[|\{p\cdot q : p,q\in P\}|\gtrsim |P|^{\frac2 3 + \frac{7}{1425}}.\]